Homogeneous control field - theory and practice, part I

Welcome to the first of a series of articles that will be published on OneRing Plus one by one at unspecified intervals. This article series aims primarily to present the basics of theory and methodology of creating the simplest form of a homogeneous field. In the later parts more complex homogeneous fields will be introduced.

What’s a homogeneous field anyway

Homogeneous Control Field is a field that consists of exactly n layers over its entire surface.
No more and no fewer layers in any spot between the three vertices of the field. In practice it means using the right portals in the area and through gradually building smaller fields achieving the transformation of these sectional fields into one big homogeneous field over the entire target area.

Part I. – A bit of theory never killed anyone

1. Road to the first field
You’re surely familiar with the situation when you layer one field after another upon a baseline… at the base you get a nice saturated color of the field, so many layers over one another that you cannot even see the roads on the map through them. The further you get from the base, the fewer fields there are above one another and the color gets translucent. We use it often. But… it’s not perfect, it doesn’t have the same color – the same number of layers – over the whole area … it’s non-homogeneous.
Non-homogeneous layering of fields upon a baseline
How to achieve homogeneousness then? Let’s go back one layer in this demonstration and have 2 layers above one another. Now we link the vertices of both fields together. This link is called a jet.
A jet is a link going out of a vertex into one of inner portals and usually closes up at least one field.
We all know this situation well, too, and we know it from the theory of efficient linking. Not only do we get the desired effect of 4 fields over 4 portals, but at the same time we achieve a situation when the whole area enclosed in the perimeter of the largest triangle consists of exactly two layers. Depending on the number of these layers we speak of the so-called field level.
Field level (order) is an expression of the number of homogeneous layers in the perimeter of the largest triangle of the field.
Second level homogeneous field (L2 HCF)

2. Upgrading the field

So now we have two layers. How to make it three? Try the same approach?

Non-homogeneous field

Nope, this is not the way. We’re missing a layer at a vertex of the field. Nevertheless the example hints at what needs to be done. This missing layer can only be gained by adding the right portals and, if necessary, changing the field vertex portal.

If we want to get an L3 HCFT, we must build auxiliary fields, called wings, above both sides of the base L2 HCF. These wings will help us achieve the necessary numbers of layers over the field’s entire surface when closing the whole field from the last vertex.

Level 3 homogeneous field (L3 HCF)

Wing is a field or a structured set of fields built on the sides above the level n-1 field.

3. Structure of a homogeneous field

Notice that the base homogeneous field is one level lower than the intended target field. Not only that, both the wings also form a field one level lower.
Homogeneous field structure
Therefore we can deduce a subsidiary theorem about the homogeneous field structure.
Homogeneous field structure theorem
Every homogeneous field of nth level consists of three homogeneous fields of n-1th level.
Analogously we can proceed to achieve higher levels and this can be considered the basic concept of the construction of a homogeneous field of theoretically any level whatsoever.

4. Portal network

If we forget about the field, the portals and links we are left with form the so-called portal network of the homogeneous field. This simplified network, which also disregards the link orientation, is used primarily as the first step in constructing a field over actual portals out in the field. The portal network concept also allows us to grasp the whole subject matter and comprehend its fundamentals.
Portal network is an organized set of portals and unoriented links of the homogeneous field.
Using the portal network and keeping in mind the previously mentioned information of homogeneous field structure we can now easily demonstrate how to create the basis of a level 4 homogeneous field (L4 HCF) using the so-called triangle method.
Creating portal network using the triangle method.

5. Observation

Did you notice the word “unoriented” in the last definition? It is important. Unoriented means lacking a designated direction. You don’t get a homogeneous field by randomly linking portals in the portal network, it is necessary to stick to a specific order. But we’ll get to that in the next part of this series. :blue_heart::vulcan_salute:
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